Finite unions of balls in C^n are rationally convex

It is shown that the rational convexity of any finite union of disjoint closed balls in C^n follows easily from the results of Duval and Sibony.Comment: V.2 - minor edits, 2 page

Uniformization of strictly pseudoconvex domains

It is shown that two strictly pseudoconvex Stein domains with real analytic boundaries have biholomorphic universal coverings provided that their boundaries are locally biholomorphically equivalent. This statement can be regarded as a higher dimensional analogue of the Riemann uniformization theorem

Solving Variational Inequalities with Monotone Operators on Domains Given by Linear Minimization Oracles

The standard algorithms for solving large-scale convex-concave saddle point problems, or, more generally, variational inequalities with monotone operators, are proximal type algorithms which at every iteration need to compute a prox-mapping, that is, to minimize over problem's domain $X$ the sum of a linear form and the specific convex distance-generating function underlying the algorithms in question. Relative computational simplicity of prox-mappings, which is the standard requirement when implementing proximal algorithms, clearly implies the possibility to equip $X$ with a relatively computationally cheap Linear Minimization Oracle (LMO) able to minimize over $X$ linear forms. There are, however, important situations where a cheap LMO indeed is available, but where no proximal setup with easy-to-compute prox-mappings is known. This fact motivates our goal in this paper, which is to develop techniques for solving variational inequalities with monotone operators on domains given by Linear Minimization Oracles. The techniques we develope can be viewed as a substantial extension of the proposed in [5] method of nonsmooth convex minimization over an LMO-represented domain

On detecting harmonic oscillations

In this paper, we focus on the following testing problem: assume that we are given observations of a real-valued signal along the grid $0,1,\ldots,N-1$, corrupted by white Gaussian noise. We want to distinguish between two hypotheses: (a) the signal is a nuisance - a linear combination of $d_n$ harmonic oscillations of known frequencies, and (b) signal is the sum of a nuisance and a linear combination of a given number $d_s$ of harmonic oscillations with unknown frequencies, and such that the distance (measured in the uniform norm on the grid) between the signal and the set of nuisances is at least $\rho>0$. We propose a computationally efficient test for distinguishing between (a) and (b) and show that its "resolution" (the smallest value of $\rho$ for which (a) and (b) are distinguished with a given confidence $1-\alpha$) is $\mathrm{O}(\sqrt{\ln(N/\alpha)/N})$, with the hidden factor depending solely on $d_n$ and $d_s$ and independent of the frequencies in question. We show that this resolution, up to a factor which is polynomial in $d_n,d_s$ and logarithmic in $N$, is the best possible under circumstances. We further extend the outlined results to the case of nuisances and signals close to linear combinations of harmonic oscillations, and provide illustrative numerical results.Comment: Published at http://dx.doi.org/10.3150/14-BEJ600 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm